Spectral and transport properties of the

two-dimensional Lieb lattice

M. Nita1, B. Ostahie1,2 and A. Aldea1,3

1National Institute of Materials Physics,

POB MG-7, 77125 Bucharest-Magurele, Romania.

2Department of Physics, University of Bucharest

3Institute of Theoretical Physics, Cologne University, 50937 Cologne, Germany.

(Dated: January 10, 2013)

Abstract

The specific topology of the line centered square lattice (known also as the Lieb lattice) induces

remarkable spectral properties as the macroscopically degenerated zero energy flat band, the Dirac

cone in the low energy spectrum, and the peculiar Hofstadter-type spectrum in magnetic field. We

study here the properties of the finite Lieb lattice with periodic and vanishing boundary conditions.

We find out the behavior of the flat band induced by disorder and external magnetic and electric

fields. We show that in the confined Lieb plaquette threaded by a perpendicular magnetic flux

there are edge states with nontrivial behavior. The specific class of twisted edge states, which

have alternating chirality, are sensitive to disorder and do not support IQHE, but contribute to

the longitudinal resistance. The symmetry of the transmittance matrix in the energy range where

these states are located is revealed. The diamagnetic moments of the bulk and edge states in the

Dirac-Landau domain, and also of the flat states in crossed magnetic and electric fields are shown.

PACS numbers: 73.22.-f, 73.23.-b, 71.70.Di, 71.10.Fd

1

arX

iv:1

301.

1807

v1 [

cond

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l] 9

Jan

201

3

I. INTRODUCTION

The interest in the line centered square lattice, known as the 2D Lieb lattice, comes from

the specific properties induced by its topology. The lattice is characterized by a unit cell

containing three atoms, and a one-particle energy spectrum showing a three band structure

with electron-hole symmetry, one of the branches being flat and macroscopically degenerate.

For the infinite lattice, the three energy bands touch each other at the middle of the spectrum

(taken as the zero energy ), and the low energy spectrum exhibits a Dirac cone located at

the point Γ = (π, π) in the Brillouin zone. Except for the presence of the flat band, the

Lieb lattice shows similarities with the honeycomb lattice in what concerns both spectral

and transport properties. For instance, besides the presence of the Dirac cone, the energy

spectrum in the presence of the magnetic field shows also a double Hofstadter picture, with

the typical√B dependence of the relativistic bands on the magnetic field B [1]. The Hall

resistance of the two systems in the quantum regime behaves alike, but the step between

consecutive plateaus equals h/e2 in the Lieb case (instead of h/2e2 for graphene) because of

the presence of a single Dirac cone per BZ. An all-angle Klein transmission is proved by the

relativistic electrons in the Lieb lattice [2–4].

There are more lattices that support flat bands, however it is specific to the Lieb lattice

that the band is robust against the magnetic field, while other lattices develop dispersion

at any B 6= 0. The intrinsic spin-orbit coupling does not affect the flat band either, but

opens a gap at the touching point Γ, the Lieb system becoming in this way a quantum spin

Hall phase [3, 5]. Topological phase transitions driven by different parameters are studied

in [6, 7]. The zero-energy flat bands became a topic of intense study also for other reasons:

they may allow for the non-Abelian FQHE [8–10] or for ferromagnetic order and surface

superconductivity [11–13].

In this paper we address the properties of the finite (mesoscopic) Lieb lattice with em-

phasis on some features of the flat band and of the edge states which are specific to this

lattice. We adopt the spinless tight-binding approach, and the spectral properties are exam-

ined under both periodic and vanishing boundary conditions applied to the system described

in Fig.1. In section II we find that the zero energy flat band exists independently of the

boundary conditions. It turns out, however, that in the periodic case the band is built up

only from B- and C- orbitals, while in the other case the A-type orbitals are also involved

2

(see Fig.1). We prove this analytically by calculating the eigenfunction in both situations. In

this way we also find out that for confined systems (i.e., with vanishing boundary conditions)

the degeneracy of the flat band equals Ncell + 1 (Ncell is the number of cells of the meso-

scopic plaquette). We find in section IIIA that for a confined plaquette two levels separate

from the bunch when a perpendicular magnetic field is applied, such that the degeneracy is

reduced by 2. This is proved in a perturbative manner for the general case, however it can

be observed more easily by the use of the toy model consisting of two cells only.

Next, we study how the flat band degeneracy is lifted by disorder and by an external

electric field applied in-plane. An exotic result is that the extended states of the disordered

flat band in the presence of a magnetic field behave according to the orthogonal Wigner-

Dyson distribution although the unitary distribution is expected. When an electric field

is applied, the flat band splits in a Stark-Wannier ladder whose structure is analyzed by

calculating the diamagnetic moments of the states in crossed electric and magnetic fields.

In section IIIB we study the edge states which fill the gaps of the double Hofstadter

butterfly when the magnetic field is applied on the confined Lieb plaquette. We identify

three types of such states. The conventional edge states located between the Bloch-Landau

bands and also between Dirac-Landau bands (i.e., the relativistic range of the spectrum)

differ, as expected, in their chirality. Additionally, we detect twisted edge states situated in

the magnetic gap which protect the zero-energy band, coming in bunches and characterized

by an oscillating chirality as function of the magnetic field. The twisted edge states show

remarkable properties: surprisingly, they are not robust to disorder, as the other types of

edge states are, and does not carry transverse current (i.e., the QHE vanishes in the energy

range covered by these states). The last property comes from a specific symmetry of the

transmittance matrix which is discussed in section IV.

Finally, one has to note that the line centered square lattices are found in nature as

Cu−O2 [14] planes in cuprate superconductors, and can be engineered as an optical lattice

[3, 15].

3

II. THE TIGHT-BINDING MODEL FOR THE LIEB LATTICE : PERIODIC VER-

SUS VANISHING BOUNDARY CONDITIONS

Our aim is to point out specific aspects of the confined Lieb plaquette from the point

of view of spectral and transport properties. In order to allow for a comparison we shortly

describe also the case of the infinite system, with and without magnetic field, although the

eigenvalue problem is already known from the literature. We remind that the continuous

model for the infinite Lieb system in perpendicular magnetic field [3], shows the√B depen-

dence on the magnetic field of the eigenenergies in the relativistic range. The information

obtained in the long-wave approximation of the Schrodinger equation, concerning the de-

pendence on B of the Bloch-Landau or Dirac-Landau bands are recaptured in the spectrum

of the discrete tight-binding model (Fig.5a) together with effects coming from the periodic

lattice and finite edges.

In this section, starting from the tight-binding Hamiltonian, we built up the eigenfunction

of the periodic and finite Lieb plaquette, and prove the degeneracy and structure of the zero-

energy flat band. The crossover from the simple Hofstadter spectrum of the simple square

lattice to the Lieb spectrum characterized by a double butterfly, magnetic gap and a flat

band is shown in Fig.3.

txyt

A B A B

A

C

C

(n+1)a (n+2)ana

ma

(m+1)a

X

Y

O

(n+1,m+1)(n,m+1) (n+2,m+1)

(n+2,m)(n+1,m)(n,m)

FIG. 1: (Color online) The Lieb lattice: the unit cell contains three atoms A,B and C; indices

(n,m) identify the cell; tx, ty are the hopping integrals along the directions Ox and Oy, respectively;

a is the lattice constant.

The Lieb lattice is a 2D square lattice with centered lines as shown in Fig.1. It is

characterized by three atoms (A,B,C) per unit cell, the connectivity of the atom A being

4

equal to four, while the connectivity of atoms B and C equals two.

Introducing creation a†nm, b†nm, c†nm and annihilation anm, bnm, cnm operators of the

localized states |Anm >, |Bnm >, |Cnm > (where (nm) stands for the cell index and the

letters A,B,C identify the type of atom), the spinless tight-binding Hamiltonian of the

Lieb lattice in perpendicular magnetic field reads:

H =∑

nmEaa†nmanm + Ebb

†nmbnm + Ecc

†nmcnm

+txe−iπmφa†nmbnm + txe

iπmφa†nmbn−1,m + tya†nmcnm + tya

†nmcn,m−1

+txe−iπmφb†nman+1,m + txe

iπmφb†nmanm + tyc†nmanm + tyc

†nmanm+1, (1)

where φ is the flux through the unit cell of the Lieb lattice measured in quantum flux units;

we mention that the vector potential has been chosen as ~A = (−By, 0, 0).

The presence of a spectral flat band can be noticed already in the simplest case of the

periodic boundary conditions and vanishing magnetic flux. Assuming that the lattice is

composed of Nxcell = N cells along Ox and Ny

cell = M cells along Oy, the Fourier transform

c~k = ckx,ky = 1√NM

∑n,m cnme

i(kxn+kym) (and similarly for all the other operators) yields the

k-representation of the Hamiltonian described by a 3× 3 matrix:

H =∑~k

(a†~k b†~k c†~k

)Ea ∆∗(kx) Λ∗(ky)

∆(kx) Eb 0

Λ(ky) 0 Ec

a~k

b~k

c~k

, (2)

where kx = 2πp/N (p = 1, .., N), ky = 2πq/M (q = 1, ..,M), and the notations ∆(kx) =

tx(1 + eikx) , Λ(ky) = ty(1 + eiky) has been used. With the choice Ea = Eb = Ec = 0, one

obtains the following eigenvalues:

Ω±(~k) = ±√|∆|2 + |Λ|2 = ±2

√t2xcos

2(kx/2) + t2ycos2(ky/2),

Ω0(~k) = 0, (3)

where Ω± are the energies of the upper and lower band, respectively, and Ω0 is the nondis-

persive (flat) band of the Lieb lattice. The most interesting point in the BZ is the point

Γ = (π, π), where in the case of the infinite lattice the three branches are touching each

other. The expansion of the functions ∆(kx) and Λ(ky) about this point gives rise to a Dirac

cone (massless) spectrum:

Ω± = ±√t2xk

2x + t2yk

2y. (4)

5

On the other hand, the expansion of the same functions about R = (0, 0) shows a parabolic

dependence:

Ω± = ±( k2x

2mx

+k2y

2my

), (5)

where mx,my are effective masses along the two directions.

Other relevant points in the BZ are M = (π, 0) and (0, π), which prove to be saddle

points in the spectrum as it can be noticed also in Fig.2. Above and below the corresponding

energy E = ±2t (where we considered tx = ty = t) the effective mass exhibits opposite signs

inducing the change of sign of the Hall effect which is visible in Fig.15.

For comparison’s sake, we remind that the energy spectrum of the honeycomb lattice

contains two cones per BZ, and that the saddle point occurs at the energy E = ±t. The

tight-binding spectrum of the graphene extends over the interval [-3t,3t], while for the Lieb

lattice the interval is [−2√

2t, 2√

2t].

FIG. 2: (Color online) The energy spectrum of the infinite Lieb lattice. (left) The case Ea =

Eb = Ec = 0 when the three bands (two dispersive and one flat) get in contact at ~k = (π, π).

At low energy the dispersion is linear giving rise to Dirac cones. (right) The staggered case

Ea = 0, Eb = Ec = 1 when the spectrum is gapped and the rounding of cones is obvious.

In order to get supplementary information about the origin of the flat band let us consider

the staggered case Ea = 0, Eb = Ec = E0. Then, a gap is expected in the energy spectrum,

6

and, indeed, the eigenvalues are now [4]:

Ω±(~k) =1

2

[E0 ±

√E2

0 + 4(|∆|2 + |Λ|2)], Ω0(~k) = E0. (6)

The new spectrum is shown in Fig.2b, where one notices the persistence of the flat band,

which is however shifted to E = E0. Since the energy E = E0 corresponds to the atomic

level of the orbitals B and C, the result argues that the flat band states are created only

by this type of orbitals. The gap induced by the staggered arrangement is accompanied by

the rounding of the cones, that indicates a non-zero effective mass in the low-energy range

of the two spectral branches, as it can be observed in Fig.2b.

In what follows we shall calculate the eigenfunctions of the finite Lieb lattice, imposing

first periodic conditions, and then the vanishing boundary conditions proper to the confined

plaquette. Let Ψ~k be the eigenfunctions of the Lieb lattice with periodic boundary conditions

built up as the linear combination:

Ψ~k = α~ka†~k|0 > +β~kb

†~k|0 > +γ~kc

†~k|0 >, (7)

where the coefficients α~k, β~k, γ~k satisfy the equations:

Eaα~k+ ∆∗(kx)β~k +Λ∗(ky)γ~k = Eα~k

∆(kx)α~k +Ebβ~k = Eβ~k

Λ(kx)α~k +Ecγ~k = Eγ~k. (8)

Then, the functions corresponding to the eigenvalues Ω0 and Ω± in Eq.(3) read:

Ψ0(~k) =1√

|∆|2 + |Λ|2(Λ∗(ky)b

†~k−∆∗(kx)c

†~k

)|0 > (9)

Ψ±(~k) =1

2

(± a†~k +

∆(kx)√|∆|2|+ |Λ|2

b†~k +Λ(ky)√|∆|2 + |Λ|2

c†~k

)|0 > . (10)

In the case of periodic conditions applied to the finite plaquette there are some subtleties

concerning the band degeneracy which become unimportant in the limit of infinite system.

It is obvious from Eqs.(9-10) that the three bands come into contact at ~k = (π, π), however

this value of ~k is allowed only if both N and M are even. In this case the flat band at E = 0

is (Ncell + 2)- fold degenerate, otherwise all the three bands are Ncell-fold degenerate (where

the number of cells Ncell = NM).

7

The expression of Ψ0(~k) in Eq.(9) indicates again that the flat band of the periodic lattice

is composed only from orbitals of the type B and C. On the other hand, we shall see below

that in the case of vanishing boundary conditions the zero-energy eigenfunction may sit also

on the A−type sites, and that the degeneracy of the flat band becomes Ncell + 1.

The periodic boundary conditions can be used in the presence of a uniform perpendicular

magnetic field for rational values of the magnetic flux φ = p/q resulting in a spectrum

composed of two Hofstadter butterflies similar to the case of the honeycomb lattice. However,

in contradistinction to the honeycomb lattice, one notices the presence of a dispersionless

band at E = 0, which is flat with respect to the variation of the magnetic flux, and is

protected by a gap opened at B 6= 0 [2, 3] . The spectrum exhibits Bloch-Landau bands at

the extremities and also relativistic Dirac-Landau bands towards the middle. The two types

of bands are distinguished by opposite chirality dE/dφ and by different dependence on the

magnetic field.

The periodic boundary conditions discussed above can be properly used for describing

infinite lattices, however when interested in mesoscopic plaquettes they have to be replaced

with vanishing boundary conditions. We intend to identify the differences introduced by the

finite size, which will turn out to be non-trivial in the case of the Lieb lattice.

For the confined Lieb lattice, the eigenfunctions can be obtained as combinations of

functions Eq.(9) or Eq.(10) with coefficients that ensure the vanishing of the eigenfunction

along the edges. As a technical detail we mention that (along the Ox direction, for instance)

the finite plaquette begins with the atom A in the first cell, and also ends with an atom A

which belong to the (N+1)-th cell. This means that the wave function |Φ(~k) > should vanish

at the site B in the 0-th and (N + 1)-th cell, i.e.: < Φ(~k)|b†N+1,m|0 >=< Φ(~k)|b†0,m|0 >= 0.

Similarly, the vanishing condition along Oy occurs at the site C in the 0-th and (M + 1)-th

cell along this direction, i.e.: < Φ(~k)|c†n,M+1|0 >=< Φ(~k)|c†n,0|0 >= 0.

In the localized representation, which is the proper one in the case of confined systems,

the eigenfunctions |Φ0(~k) > corresponding to E = Ω0 = 0 look as follows:

|Φ0(~k) >=

√2

N + 1

√2

M + 1

N+1∑n=1

M+1∑m=1

( 2tycosky2√

|∆|2 + |Λ|2sinkxn sinky(m−

1

2) b†nm|0 >

−2txcos

kx2√

|∆|2 + |Λ|2sinkx(n−

1

2) sinkym c†nm|0 >

), (11)

where kx, ky are obtained from the condition that the wave function vanishes at the boundary,

8

and equal kx = pπ/(N+1) (p=1,..,N+1) and ky = qπM+1

(q=1,..,M+1). Since the situations

p = N + 1 (at any q) and q = M + 1 (at any p), generate |Φ0 >= 0, we are left in Eq.(11)

with only NM non-vanishing degenerate orthogonal eigenfunction.

The eigenfunctions |Φ±(~k) > corresponding to the other two energy branches can be

written similarly as:

|Φ±(~k) >=

√2

(N + 1)(M + 1)

N+1∑n=1

M+1∑m=1

(± sinkx(n−

1

2)sinky(m−

1

2) a†nm|0 >

+2txcos

kx2√

|∆|2 + |Λ|2sinkxn sinky(m−

1

2) b†nm|0 >

+2tycos

ky2√

|∆|2 + |Λ|2sinkx(n−

1

2)sinkym c†nm|0 >

), (12)

where states of the type A are this time also present. One can readily see that the number of

non-vanishing states in each spectral branch is (N+1)(M+1)−1, since the point Γ = (π, π)

has to be treated separately. This is because its corresponding energy vanishes and the state

should be counted in the flat band. In this case the wave function becomes:

|Φ0a >=: |Φ±(π, π) >=

√1

N + 1

√1

M + 1

N+1∑n=1

M+1∑m=1

(−1)n+m a†nm|0 > . (13)

For the finite Lieb plaquette with vanishing boundary conditions, one may conclude that the

flat band degeneracy equals NM+1, while each other branch contains NM+N+M states,

so that the total number of states equals indeed the number of sites 3NM + 2(N +M) + 1.

In the presence of the magnetic field, the vanishing boundary conditions give rise to edge

states which fill the gaps of the Hofstadter spectrum corresponding to the periodic system.

Besides the edge states existing in the energy range of the Bloch-Landau levels (which are

the only met for the finite plaquette with simple square structure), there are edge states

in the relativistic range which show opposite chirality [16], but also non-conventional edge

states lying in the central gap which protects the zero energy dispersionless band. This

last new class of edge states exhibits oscillating chirality when changing either the magnetic

flux or the Fermi energy. These states will be studied in the next chapter. The fate of the

zero-energy states in the presence of confinement will be discussed in the next section.

The Lieb lattice can be generated from the simple square lattice by extracting each the

second atom when moving along both Ox and Oy direction. Formally, this means either

to push to infinity the energy Ed of these atoms or to cut down the hopping integrals t′

9

connecting them to the neighboring atoms, and it is instructive to follow the change of the

spectrum when Ed/t′ →∞. By driving the system in this way from 1 to 3 atoms/unit cell,

the lattice periodicity is doubled along both directions, and the flat band is generated. The

middle panel of Fig.3 shows how the butterfly wings break off during the process giving rise

to the relativistic range.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-4

-3

-2

-1

0

1

2

3

4

Eig

enva

lues

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-3

-2

-1

0

1

2

3

Eig

enva

lues

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-2.4

-1.6

-0.8

0.0

0.8

1.6

2.4

Eig

enva

lues

FIG. 3: The energy spectrum as function of the magnetic flux for three values of the hopping

integral t′ (see text) : (left) t′ = 1, corresponding to the simple square lattice, (middle) t′ = 0.5,

(right) t′ = 0, corresponding to the square lattice with centered lines (Lieb lattice). φ is the

magnetic flux through the unit cell of the simple lattice measured in quantum flux units. The

Hofstadter butterfly is obvious for t′ = 1, while a doubled butterfly results for t′ = 0 in each of the

intervals φ ∈ [0, 0.25], φ ∈ [0.25, 0.5], etc. (one has to keep in mind that the flux through the Lieb

unit cell is four times larger than φ). The energy is measured in units of hopping integral t.

III. SPECIFIC ASPECTS OF THE FINITE LIEB PLAQUETTE IN MAGNETIC

FIELD: ZERO ENERGY FLAT BAND AND TWISTED EDGE STATES

A. The properties of the flat band

There are several pertinent questions which can be asked concerning the flat band in the

energy spectrum of the Lieb finite system: what is the degeneracy, what is the response to

the magnetic and electric field and to the disorder?

Let us find first the conditions which should be satisfied by the zero energy eigenfunction

ΨE=0. Let H be the tight-binding Hamiltonian of a finite system and ΨE its eigenfunctions:

H =∑n

En|n >< n|+∑n,m

tnm|n >< m|, ΨE =∑n

αn|n >, (14)

10

where |n > is a basis of functions localized at the sites n. The condition HΨE = 0

generates a set of equations for the coefficients αn:

Enαn +∑m

tnmαm = 0, ∀n. (15)

Eqs.(15) are the necessary and sufficient conditions which must be fulfilled by the wave

function ΨE in order to correspond to the zero eigenvalue E = 0. With En = 0, and

taking into account only the nearest neighbors (tnm = t) the above equations become simply∑m∈Vn αm = 0, for any n, where the sum is taken over all sites in the first vicinity Vn of the

site n. In addition, if ΨiE=0 and Ψj

E=0 are two degenerate states , the orthogonality condition

reads∑

n αinα

jn = 0. The number of configurations αn which satisfy simultaneously the

two conditions equals the dimension of an orthogonal basis in the space of the degenerate

eigenfunctions at E = 0.

1 1

0

0 0

1

10 1

−11

10 −1 0

−1

000 1−1 0010

−2

00 −1

−1 0 01 −1

−1

0

00 0

(a) (b) (c)

FIG. 4: (Color online) The three eigenstates of the flat band for a Lieb lattice composed of two

cells. The eigenfunctions are Ψ(0) =∑

nm αnm|nm > and the coefficients αnm are indicated. We

notice that the condition for the flat band appearance∑

nm∈Vn0m0αnm = 0 holds for any site

n0m0.

An instructive illustration is the Lieb plaquette consisting of two cells (see Fig.4). The

plaquette contains 13 atoms (6 of type A, 4 of type B and 3 of type C). There are three

configurations of the coefficients αn which satisfy the conditions discussed above and they

are pictured as (a), (b) and (c). (The numbers 0,−1, 1,−2 mentioned in Fig.4 represent

the values, up to the normalization factor, of the coefficients αn).

With the notations used in the Hamiltonian (1), the three states can be written as:

Ψ(0)1 (E = 0, φ = 0) = [−a†11 + a†21 − a

†22 + a†12 − a

†31 + a†32] |0 >

Ψ(0)2 (E = 0, φ = 0) = [b†11 − b

†21 − c

†11 + c†31 + b†12 − b

†22] |0 >

Ψ(0)3 (E = 0, φ = 0) = [b†11 + b†21 − c

†11 − 2c†21 − c

†31 + b†12 + b†22] |0 > (16)

11

It is obvious that∑

n αin = 0 for any i = 1, 2, 3 and that < Ψi|Ψj >= 0 for any i, j = 1, 2, 3,

i.e. the three states correspond to E = 0 and are mutually orthogonal.

Next, we want to find out how the zero energy states Eq.(16) respond to a perpendicular

magnetic field. In order to answer this question, we write the Hamiltonian (1) as:

H(φ) = H(0)(φ = 0) +H(1)(φ),

H(1)(φ) =∑nm

(a†nmbnm + b†nman+1,m

)(e−iπmφ − 1) +H.c., (17)

and perform degenerate perturbation with respect to H(1). Applying this approach to the

two-cell Lieb system the matrix elements involved are < Ψ(0)1 |H1|Ψ(0)

2 >= 8i sinπφ and

< Ψ(0)2 |H1|Ψ(0)

3 >= 0 and the secular equation reads:

det

−E 8i sinπφ 0

−8i sinπφ −E 0

0 0 −E

= 0,

giving rise to the eigenvalues: E1,2 = ±8tsinπφ and E3 = 0.

One remarks that the bulk state Ψ3 does not couple to the magnetic field and its eigenen-

ergy remains E3 = 0. On the other hand, the surface states Ψ1,2 get a dispersion which

depends on φ. The conclusion of the perturbative calculation is that the magnetic field

reduces by 2 the degeneracy of the zero energy band.

Let us generalize now to a finite Lieb lattice containing N cells along the Ox-axis and

M cells along Oy-axis, so that the total number of cells is Ncell = NM and the number

of states is 3NM+2(N+M)+1. It has been proved in the previous chapter that, at zero

magnetic field, the number of zero energy degenerate states is Ncell + 1. Then, the two-cell

model shows that in the presence of the magnetic field two states separates from the bunch

so that the degeneracy of the flat band becomes Ncell − 1. Using a similar approach for

the general case, one has to use the eigenfunctions Eq.(11) and Eq.(13) and the expression

Eq.(17) as the perturbation. One finds out easily that < Φ0(~k)|H(1)Φ0(~k′) >= 0, and that

the only nonvanishing matrix elements are X(~k) =:< Φ0(~k)|H(1)Φ0a >. In the general case,

12

the secular equation becomes:

det

−E 0 0 ... X(~k1)

0 −E 0 ... X(~k2)

... ... ... ... ...

X(~k1) X(~k2) X3 ... −E

= 0,

which in the polynomial form reads EN−2(E2−X2) = 0, where X2 = X2(~k1) + ...X2(~kN−1).

This formula (where N stands here for the degeneracy of the flat band) says that from the

whole bunch only two levels get a dispersion depending on φ, meaning that the degeneracy

of the zero energy level is reduced by 2 in the presence of the magnetic field. So, the general

finite Lieb plaquette behaves similarly to the two-cell model.

The numerically calculated energy spectrum of the finite plaquette in perpendicular mag-

netic field is shown in Fig.5a, where one can check again the presence of the two levels which

separates from the flat band while the most of the bunch at E = 0 consisting of Ncell − 1

states remains dispersionless.

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

Eig

en-e

nerg

ies

magnetic flux

(a)

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

Eig

en-e

nerg

ies

magnetic flux

(b)

FIG. 5: The Hofstadter-type spectrum of a finite Lieb lattice of dimension Nxcell = Ny

cell = 10. (a)

for the clean plaquette, and (b) for the disordered one (disorder strength W = 1); the flux φ is

measured in quantum flux units.

The strong degeneracy of the flat band can be however lifted by a disordered potential.

The broadening of the band depends on the strength of the disorder, however it remains

independent of the magnetic field as in the case of the clean system (see Fig.5b). We use a

diagonal disorder of the Anderson-type characterized by the width parameter W [17]. The

13

calculation of both the inverse participation number (IPN) and of the interlevel distribution

indicate that in the middle of the disordered band the states are still delocalized, and

described by the orthogonal Wigner-Dyson distribution (β = 1) which is the typical result

in the absence of the magnetic field. This proves once more the absence of response of the

flat band to the perpendicularly applied magnetic field, even in the presence of disorder.

The inverse participation number (IPN) is defined as:

IPNE =∑n

| < n|ΨE > |4 (18)

and indicates the degree of localization of the states. The small values of the IPN for

energies in the middle of the density of states denotes the presence of extended states, and,

as expected, the localization increases towards the band edges. The numerically calculated

density of states and the dependence on energy of the inverse participation number are shown

in Fig.6a. Further information about the localization and the response to the magnetic field

is provided by the distribution function of the level spacing between consecutive eigenvalues

sn = En − En−1 of the disordered system. Let us define the dimensionless quantity tn =

sn/ < sn >, where < sn > is the mean level spacing. In the disordered system, in the range of

delocalized states, the level spacings are distributed according to the Wigner-Dyson surmise

[18]:

P(t) = bβtβe−aβt

2

, (19)

where β = 2 in the presence of the magnetic field, and β = 1 if B = 0. As a signature of the

distribution, the variance of the level spacing δt =< δs > / < s > is < δt >= 0.4220 in the

first case, and < δt >= 0.5227 in the second one. Fig.6b shows the numerically calculated

variance of the level spacing distribution, and one can notice that, in the middle of the flat

band, where the states are delocalized, the variance is < δt >= 0.5227. This means that,

despite the presence of the magnetic field, the flat band behaves according to the orthogonal

(β = 1) Wigner-Dyson distribution instead of the unitary one (β = 2), as it is expected at

B 6= 0.

Another way to lift the degeneracy of the zero-energy band is to apply an in-plane static

electric field. We expect specific aspects coming from the existence of the edges and of

the lattice structure. In the numerical calculation the electric field applied along Oy-axis

is simulated by replacing the atomic energies Enm with Enm + Eyn, where yn is the site

coordinate along Oy. Fig.7a shows how the eigenvalues stemming from the flat band are

14

0

500

1000

1500

2000

2500

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0

0.05

0.1

0.15

0.2D

OS

(ar

bitr

ary

units

)

IPN

Energy

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Var

ianc

e of

lev

el s

paci

ng

Energy

0.4220(b) 0.5227

FIG. 6: (Color online)(a) The density of states and IPN in the flat band range as function of the

energy for a disordered Lieb plaquette of dimension 20×20 cells averaged over 1000 configurations

(disorder strength W = 0.3). (b) The variance of the level spacing distribution as function of

energy for the same disordered system; the horizontal lines correspond to 0.4220 (as for the unitary

ensemble) and 0.5227 (as for the orthogonal ensemble).

split in several degenerate mini-bands which develop a Stark fan with increasing electric field.

It can be checked that the number of mini-bands equals the number of lattice cells along the

direction of the electric field. A perpendicular magnetic field gives rise to supplementary

fine splitting and to the presence of states between mini-bands. This can be seen in Fig.7b

and also in Fig.8. We have noticed that the flat band states are much more sensitive to the

electric field than the edge states, and they give rise to a Wannier-Stark ladder at values of

the electric field E for which the edge states are still non affected. We have also numerically

observed that the wave function in the l− th miniband is mainly localized in the l− th row

of cells in the direction of the electric field.

We already have seen that the flat band states do not show any diamagnetic response,

and it is somehow surprising that the Wannier-Stark states coming from the former flat band

exhibit a diamgnetic moment when the magnetic field is applied. It is interesting that each

mini-band shows both positive and negative magnetic moments, and Fig.8a suggests that

the chirality dE/dφ changes the sign at the center of the mini-band. We have studied also

the localization properties of the eigenstates, particularly the localization along the edges

P edgeα , defined as:

PEdgeα =

∑i∈Edge

|Φα(i)|2, (20)

15

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5

Eig

enva

lues

Eα

(a)

Electric potential (eεL y)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5

Eig

enva

lues

Eα

(b)

Electric potential (eεL y)

FIG. 7: The low energy spectrum of a finite Lieb plaquette as function of the electric potential

applied on the plaquette in the Oy-direction at: (a) φ = 0, and (b) φ = 0.12.

-0.04

-0.02

0

0.02

0.04

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Mag

netiz

atio

n

Energy

(a)

0

0.2

0.4

0.6

0.8

1

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Edg

e Lo

caliz

atio

n

Energy

(b)

FIG. 8: The behavior of the flat band in crossed magnetic and electric field. (a) The orbital

magnetization Mα, and (b) the edge localization P edgeα (b) vs. energy Eα for a finite Lieb lattice

of dimension Nxcell = Ny

cell = 15. The flat band turns into a set of 15 minibands, every miniband

being composed of two parts with opposite magnetization. The states in the lowest and highest

miniband have significantly increased edge localization (φ = 0.12 and eELy = 0.2).

where the index α indicates the state, and the sum is taken over all the sites which belong

to the plaquette boundary. It turns out that the states which belong to the mini-bands from

the extremities of the fan spectrum are strongly localized along the edges (Fig.8b). The

localization is of electric origin since the picture is similar no matter whether the magnetic

field is present or not.

We conclude, saying that the disorder lifts the degeneracy of the flat band keeping the

states independent of the magnetic field, while the electric field produces states which re-

16

spond to the magnetic field and show specific diamagnetic moments.

B. The twisted and type II edge states and their properties

The confinement of the Lieb lattice induces several types of edge states. Besides the

conventional edge states found in the Bloch-Landau and Dirac-Landau regions, there are

still two other classes of edge states. We discuss first the twisted edge states lying in the

magnetic gap opened around the degenerated energy level E = 0. Although the new states

are localized along the perimeter of the plaquette, they do not follow the known behavior

of the conventional edge states. The new class of edge states manifest specific properties: i)

their energy depends on the flux in a periodic way. This means that the chirality defined by

the sign of dE/dφ is not conserved but alternate when changing the flux, in contradistinction

to the usual edge states either in the Bloch-Landau or Dirac-Landau domain. Obviously, the

alternate chirality should be reflected also in oscillations of the orbital magnetization at the

variation of the magnetic flux. ii) their energies as function of the flux appear as twisted into

bunches; for the clean square plaquette shown in Fig.9a each bunch consists of four states.

iii) the states prove the lack of robustness against disorder and iv) prove specific transport

properties, namely, the twisted edge states carry a finite longitudinal resistance accompanied

by vanishing Hall resistance. A piece of the spectrum of the clean plaquette in the energy

range of twisted edge states is shown in Fig.9a, where bunches consisting of four twisted

edge states can be observed. One also has to notice that, at a given flux, the states in the

bunch may show opposite chirality meaning that they carry diamagnetic currents moving

in opposite directions. In the presence of disorder (Fig.9b) one notices that the twisted

eigenenergies get stretched but the rest of the spectrum (the band and the edge states in

the Dirac region) is not affected. This indicates that the twisted states are very sensitive

to disorder. The understanding of this effect is simple in the sense that the degeneracy at

crossing points [19] is lifted by the perturbation introduced by the impurity potential, and

this occurs even at weak disorder. The Lieb lattice exhibits still another specific edge states

(which we call type-II edge states), which in Fig.12 are placed immediately above the Dirac-

Landau bands at the transition from Dirac bulk to conventional edge states. They cannot be

identified according to the sign of the magnetic moment [20] since their chirality dEn/dφ is

the same as for the bulk (band) states [16]. Nevertheless, the diamagnetic currents of these

17

-1

-0.9

-0.8

-0.7

-0.6

-0.5

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

Eig

enen

ergi

es(a)

magnetic flux

-1

-0.9

-0.8

-0.7

-0.6

-0.5

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

Eig

enen

ergi

es

(b)

magnetic flux

FIG. 9: The eigenenergy in the range of the twisted edge states vs. the magnetic flux φ for a

pure Lieb lattice (a) and for a disorder Lieb lattice (b). The twisted edge states has an oscillatory

behavior when the magnetic flux is varied, and they form bunches with four states in each bunch.

The oscillatory behavior is destroyed by disorder in the right figure, but the conventional edge

states (shown in the lower part of the spectrum) remain robust against disorder. The dimension

of the Lieb lattice is Nxcell = Ny

cell = 10 and the amplitude of the Anderson disorder is W=1.

(a) (b)

FIG. 10: The behavior of the edge states with disorder. (a) The absence of the backscattering for

a conventional edge states. A pair of twisted states which are sufficiently close in energy may suffer

the backscattering suggested in (b), which induces the localization shown in Fig.11.

states are located along the edges of the plaquette. These edge states show a double-ridge

profile and carry current in both directions, but nonetheless the total magnetization remains

of bulk-type.

In Fig.13 the diamagnetic currents of bulk states, type-II edge states and of conventional

edge states are sketched. The twisted edge states may show currents similar to both con-

ventional and type-II edge states. Compared to the twisted states, the type-II edge states

behave substantially different in the electronic transport. These states will be studied in de-

tail elsewhere. The contribution to the magnetization of each eigenstates |α > is calculated

18

FIG. 11: |Ψ|2 calculated for a conventional edge state in the Dirac range (left) and for a twisted

edge state (right) for a disordered plaquette with W = 0.2. One observes that this low disorder

does not affect the conventional edge state but localizes the twisted state.

-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1

-0.9

-0.8

-0.7

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

Eig

enen

ergi

es

magnetic flux

twisted edge states

conventional edge states

type-II edge states

FIG. 12: The eigenenergies vs. the magnetic flux φ for a pure Lieb lattice in the range which

emphasizes the type II edge states. In the spectrum, they are located between the bulk states in

the Dirac-Landau range and the conventional edge states of the first gap. Their energy decreases

with the magnetic field similar to the bulk states, however they have edge localization of the wave

function. The dimension of the Lieb lattice is Nxcell = Ny

cell = 10.

19

(a) (b) (c)

FIG. 13: The sketch of the diamagnetic currents in the Dirac-Landau range of the spectrum: (a)

the counterclockwise loop of a bulk state, (b) the double ridge current of a type-II edge states, and

(c) the clockwise loop of a conventional edge states. The twisted edge states may show both (b)

and (c) aspect.

following the approach from [21], namely:

Mα = − < α|dHdφ|α >

= iπtx∑mn

(me−imπφ < α|Anm >< Bnm|α > −meimπφ < α|Anm >< Bn−1,m|α >

+ me−imπφ < α|Bnm >< An+1,m|α > −meimπφ < α|Bnm >< Anm|α >). (21)

All the matrix elements in the above equation are known once the eigenstates |α > are

calculated numerically in the presence of the magnetic flux. Fig.14 depicts the diamagnetic

moments and the localization at the edges of different type of states. The bulk (band) states

show positive magnetization and vanishing localization at the edge, the conventional edge

states show negative magnetization, and 97% localization at the edge. The twisted edge

states show alternating magnetization, as expected, but also an unanticipated differences

in the degree of edge localization. This is because they exhibit either a single- or double-

ridge wave function. Obviously, the double-ridge wave function is not strictly stuck to the

edge so that Pedge ≈ 0.7− 0.8, while for the single-ridge states the same parameter goes up

to 0.9. Fig.14 points out that the single-ridge states which are localized close to the edge

exhibits negative magnetization (as the conventional edge states), while the double-ridge

states exhibits positive magnetization.

IV. THE INTEGER QUANTUM HALL EFFECT

The quantum transport of the 2D Lieb plaquette shows some similarities with the case of

graphene, however it also reveals particular properties. The Hall resistance as function of the

20

-20

-10

0

10

20

Mα

Twistededge states

Type-IIedge states

Conventional

(a)

edge states

0

0.2

0.4

0.6

0.8

-1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7

Ped

ge

Eα

(b)

1

FIG. 14: (Color online) Magnetization Mα and localization at the edges Pedge corresponding to

the eigenenergies Eα: the bulk (band) states (green) show positive magnetization and vanishing

localization at the edges; the conventional edge states (blue) show negative magnetization and 97%

localization at the edges; the type-II and the twisted states show 60−80% localization at the edges.

The data are for a clean Lieb plaquette of dimension Nxcell = Ny

cell = 10 and φ = 0.16.

Fermi energy at a given quantizing magnetic field was obtained in Ref.3 by calculating the

Chern numbers, and has the general aspect which can be observed in Fig.15 which we obtain

in the Landauer-Buttiker formalism: starting from the bottom of the spectrum, RH shows

h/e2 steps in the Bloch-Landau region, then change the sign, and shows again h/e2 steps in

the Dirac-Landau region. The steps of the quantum Hall plateaus differs from those of the

honeycomb lattice since in the Lieb case there is only one Dirac cone per the unit cell. The

change of sign is associated with the opposite chirality of the edge states in the two regions

and occurs at E = ±2t, while in graphene the same change takes place at E = ±t [22]. The

density of states (shown in blue in Fig.15) is calculated as DOS = − 1π

∑n ImG

+nn(E), where

21

G+ is the retarded Green function for the mesoscopic plaquette connected to the leads. In

order to calculate the transport properties the mesoscopic plaquette must be connected to

leads, the whole system being described in the tight-binding approach by the Hamiltonian:

H = HS +HL + τHLS, (22)

where the first term is just the Hamiltonian (1), the second term describes all the leads, and

HLS couples the leads to the plaquette with the strength τ . With G+α,β(E) ≡< α|(E −H +

i0)−1|β >, the electron transmittance between the leads α and β, in the Landauer-Buttiker

formalism, is given by:

Tα,β = 4τ 4|G+α,β(EF )|2ImgLα(EF )ImgLβ (EF ), (23)

where gL is the Green function of the isolated leads. In what follows we shall discuss the

interesting question of the contribution to transport of the twisted edge states introduced in

the previous section. The answer can be found in Fig.15 in the energy range E ∈ [−0.8,−0.6],

where one observes that the twisted edge states found in that range do not support the Hall

resistance (RH = 0), however they contribute to the longitudinal resistance, which exhibits

an oscillating behavior. It is also to notice that all the oscillations minima equals RL = 1/4,

a fact that should find its explanation.

In exploring these unexpected effects, the first step should be to identify the transmittance

matrix. The numerical investigation presented in Fig.17a shows that, in the range of the

twisted edge states the properties of transmittances Tα,β are very specific: they are not

quantized, show an oscillating dependence on the energy, and, mainly, satisfy the symmetry

relation:

Tα,α+1 = Tα+1,α, (24)

while in the range of the conventional edge states, where the quantized plateaus occur, the

usual properties of quantum Hall effect hold: Tα,α+1=integer and Tα+1,α = 0 (for a given

sign of the magnetic flux). Combining Eq.(24) with the general property∑

α Tα,β = 0, which

expresses the current conservation, it turns out that the transmittance matrix for the edge

22

-1

-0.5

0

0.5

1

1.5

2

-2.5 -2 -1.5 -1

RH

, RL,

DO

S

Energy

RH

DOS

RL

RL

FIG. 15: (Color online) The transport properties of the Lieb lattice under the magnetic field:

Hall resistance RH , longitudinal resistance RL and density of states DOS for a finite Lieb lattice

connected to four transport leads. The quantum Hall effect can be observed for E ∈ [−2.75,−0.8]

(RH integer and RL = 0). In the Bloch-Landau part of the spectrum (E < −2) one has RH < 0,

while in the Dirac-Landau part (E > −2) one has RH > 0. In the energy range E ∈ [−0.8,−0.6]

the transport properties are due to the twisted edge states and we get zero Hall resistance RH = 0

and oscillations of the longitudinal resistance with the characteristic minima at RL = 1/4. The

density of states exhibits maxima at the transition between the Hall plateaus and for the energy

values where the twisted edge states appear. The dimension of the plaquette is 10× 30 unit cells,

the magnetic flux is φ = 0.12. The resistance is in units h/e2, DOS in arbitrary units, and the

energy in units t.

transport in the domain of twisted edge states can be written as:

T =

−2T T 0 T

T −2T T 0

0 T −2T T

T 0 T −2T

.

The transmittance Tα,β relates the current through the lead α to potentials at the contact

sites β as:

Iα =e2

h

∑β

TαβVβ, (25)

and, for the four-lead device considered in Fig.16, in the Landauer-Buttiker formalism, the

23

4

2

31

FIG. 16: (Color online) The four-lead Hall device: illustration of the edge currents carried by the

twisted edge states, for which the symmetry relation Tα,α+1 = Tα+1,α holds.

Hall and longitudinal resistance are given (in units h/e2) by:

RL = R14,23 = (T24T31 − T21T34)/D,

RH = (R13,24 −R24,13)/2 = (T23T41 − T21T43 − T32T14 + T12T34)/2D, (26)

where D = −4T 3 is a subdeterminant of the matrix T . By the use of the above equations

and of transmittance matrix T , valid in the range of twisted edge states, one obtains im-

mediately, a vanishing Hall resistance (RH = 0) and the longitudinal resistance RL = 1/4T

(in units h/e2). The minima of RL observed in Fig.17b correspond to the maxima of the

transmittance, and, obviously, the value RL = 0.25h/e2 expresses a perfect conducting one

channel transport with T = 1. It turns out that, although carried by edge states, the current

shows a dissipative character. The oscillations of T12 and T21 in the domain of the twisted

edge states (E ∈ [−0.95,−0.75]) follow the similar oscillations of the density of states, while

in the quantum Hall regime (E ∈ [-1.5,-0.95]) the DOS is flat.

Another interesting problem is the transition between the first and second plateau of the

Dirac-Landau region (E ∈ [−1.55,−1.5]) which is much wider than similar transitions in the

Bloch-Landau region. The transition get a width which is due to the presence of the type-II

edge states (observed in the spectrum Fig.12 above the Dirac band), and is accompanied by

oscillations of the transmittance (see T12 and T21 in Fig.17a). We have to stress that in this

energy range T12 and T21 are no more equal, and the numerical calculation suggests that

T12 = T21 + 1. It means that the symmetry Eq.(24) remains specific to the twisted edge

states.

24

0

0.5

1

1.5

2

-1.6 -1.4 -1.2 -1 -0.8 -0.6

T12

, T

21

Energy

(a)

T12

T21

0

0.5

1

1.5

2

-1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65

RH

, RL,

T12

Energy

(b)

RL

T12RH

FIG. 17: (Color online) (a) The transmittances T12 and T21 showing the quantized values T12 = 1

and T21 = 0 in the range of the IQHE, non-quantized oscillating values T12 = T21 in the range

of twisted edge states E > −0.95, and T12 = T21 + 1 in the range of type-II edge states E ∈

[−1.55,−1.5]. (b) The Hall and longitudinal resistance: RH (blue line) vanishes in the range of

twisted edge states, while RL (red line) exhibits oscillations with minima RL = 0.25. The minima

of the longitudinal resistance occur at the energies where the transmittance (black dashed line) get

the maximum value T12 = T21 = 1. The dimension of the plaquette is 10× 30 unit cells, φ = 0.16.

In order to figure out a scenario for the vanishing Hall effect, we remind first that, in

the range of the QH effect, all the (conventional) edge states responsible for the plateaus

of the transverse magnetoresistance get a unique chirality determined by the direction of

the magnetic flux. This results in a definite sense (say clock-wise) of the current such that

Tα,α+1= integer and Tα+1,α = 0. The symmetry Tα,α+1 = Tα+1,α which occurs in the range

of the twisted edge states is characteristic to the absence of the magnetic field. So, this

symmetry indicates a ’loss of influence’ of the magnetic field followed by a vanishing Hall

effect. As a support of this idea we note that the twisted edge states show alternating (clock

and anti-clock) chiralities which allow for the transmittance in both directions, as sketched in

Fig 16. This might be an heuristic explanation for the vanishing of the transverse resistance

despite the presence of edge states.

Let us discuss now the effect of the disorder. It is known that the conventional edge

states are robust to disorder, whereas the bulk states (which form the Landau bands) are

more sensitive, so that the IQHE of a disordered plaquette shows robust plateaus, and

a broadened transition region between two consecutive plateaus. On the other hand, as

we have shown, the disorder localizes easily the twisted edge states, changing in this way

25

0

0.2

0.4

0.6

0.8

1

-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6

RH

, T12

, T21

T12

T21

RH

Energy

FIG. 18: (Color online) The Hall resistance RH (red) and transmittances T12 (green) and T21 (blue)

for a disordered Lieb lattice in the transition region from the first Hall plateau to the domain of

twisted edge states. The figure shows that the symmetry relation T12 = T21 for twisted edge

state transport holds also in the presence of disorder. The Hall resistance is RH = 1 for the

energies corresponding to the first gap with conventional edge states, and RH = 0 for the energies

corresponding to the twisted edge states. The dimension of the plaquette and the magnetic flux

are the same as in Fig.17, the disorder amplitude is W = 0.5.

their transmittance properties. Fig.18 depicts the transmittances T12, T21 for a disordered

Lieb plaquette compared to the same transmittances of the clean system. One observes the

quantized values in the range of the conventional edge states followed, in the range of twisted

states, by reduced values of the disordered transmittance which replace the peaks specific

to the clean system. It is worth to say that the symmetry Tα,α+1 = Tα+1,α is preserved for

each individual disordered sample, and, as a consequence, the Hall effect vanishes similar to

the clean case.

V. CONCLUSIONS

We have found that the specific topology of the 2D Lieb lattice induces remarkable

spectral and transport properties. Up to point there are similarities with the electronic

energy spectrum of graphene in what concerns the presence of a Dirac-type cone at low

energy, however, in addition, a macroscopically degenerated flat band occurs at the middle

of the spectrum. The perpendicular magnetic field applied on a finite (mesoscopic) Lieb

26

plaquette opens a gap around the flat band, and we show the presence in this gap of a

new class of edge states with alternating chirality (which we call twisted edge states). The

flat band is insensitive to the magnetic field, however an in-plane electric field, and also the

disorder, lifts the degeneracy. The electric field applied on a finite (mesoscopic) system gives

rise to a Wannier-Stark fan composed of degenerate mini-bands, the number of them being

equal to the number of cells along the direction of the field.

The macroscopic degeneracy of the flat band is lifted by disorder, and the degree of

localization and the level spacing distribution are studied. It turns out that not only the

ordered flat band, but also the disordered one, does not feel the magnetic field; indeed, we

prove that the level spacing distribution of the disordered system follows the orthogonal (β =

1) Wigner-Dyson distribution, which usually describes disordered systems in the absence of

the magnetic field.

We calculate analytically the orthogonal eigenfunctions of the finite Lieb system cor-

responding to the three spectral branches in the low energy range, both for periodic and

vanishing boundary conditions. In this way we find also the degeneracy of the zero energy

flat band which, in the periodic case, equals the total number of unit cells Ncell (except when

both Nxcell and Ny

cell are even numbers, in which case the degeneracy increases to Ncell + 2),

while in the case of the closed boundaries the degeneracy is Ncell+1. A toy model composed

of only two unit cells helps to understand the behavior in the presence of a perpendicular

magnetic field. The perturbative calculation shows that 2 states of the flat band separate

from the degenerated bunch, and belong to the class of twisted edge states.

The eigenenergies of the twisted edge states depend in an oscillatory manner on the

magnetic flux, i.e., show an alternating chirality, and contrary to the conventional edge

states, the diamagnetic moment change the sign when the magnetic flux is varied. These

type of edge states generated by the magnetic field are not protected by the broken time-

reversal symmetry, and proves to get localized even at low disorder, when the conventional

edge states remain robust.

The transport properties are calculated by attaching leads to the finite Lieb system and

using the Landauer-Buttiker formalism. The quantum Hall resistance looks similar to that of

the graphene except the steps are equal to h/e2 (instead of h/2e2), however in the domain of

the twisted edge states the properties become unconventional: the Hall resistance vanishes,

while the longitudinal one shows oscillations which can be correlated with the oscillations

27

of the density of states (calculated in the presence of the leads). This behavior stems from

the symmetry of the transmittance Tα,α+1 = Tα+1,α which occurs despite the presence of the

quantizing magnetic field. The symmetry holds also in the presence of disorder.

VI. ACKNOWLEDGEMENTS

We acknowledge support from PNII-ID-PCE Research Programme (grant no 0091/2011),

Core Programme (contract no.45N/2009) and Sonderforschungsbereich 608 at the Institute

of Theoretical Physics, University of Cologne. One of the authors (AA) is very much in-

debted to A.Rosch for illuminating discussions.

[1] R. Rammal, J. Physique 46, 1345 (1985).

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